---
tag: 0GMV7GP8W0
kind: statement
role: proposition
name: Absolute value is multiplicative
msc: [26A03]
uses: [00S7C68D51]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_value
    locator: Real numbers
---

## Statement

For all real numbers $a$ and $b$, $|a b| = |a| \, |b|$.
Here $|\cdot|$ is the [[absolute value|00S7C68D51]].

## Remarks

Proof: check the four cases of the signs of $a$ and $b$.
In particular $|a^n| = |a|^n$ for each natural number $n$, and $|a / b| = |a| / |b|$ for $b \neq 0$.
The [[modulus of complex numbers|0HYNZE940T]] is also multiplicative.

## Explanations

### L1 human

You can take the absolute value before or after a product, and you get the same result.
For example, $|(-2) \cdot 3| = |-6| = 6$ and $|-2| \cdot |3| = 2 \cdot 3 = 6$.